Filters are used extensively in telecommunications networks. An example is shown in FIG. 1 for an xDSL service in which HP are high pass filters, LP are low pass filters, H is a hybrid circuit, POTS is a telephone network and UP-DSL and DOWN-DSL are representations of the uplink and down link transmission lines of the xDSL system. Line drivers such as Asymmetric Digital Subscriber Line (ADSL) drivers, Very High Speed Digital Subscriber Line (VSDL) drivers and others generally called in the art xDSL line drivers (where “x” represents the type of technology) are known.
It is known to design passive filters of any order. Generally, filter designs have been characterised into various “standard” filter types generally known by their original proponent such as Chebyshev, inverse Chebyshev, Butterworth, Bessel, Cauer (elliptic). These standard designs have recognisable and useful characteritics. A design engineer will select a particular type of filter depending upon the application. Standard tables of component values for such filter designs exist. Nowadays, user-friendly computer programs are available on the Internet for providing component values as well as characteristics of a selected filter design.
The frequency response of a filter may be described by its transfer function H(ω), where ω represents 2π times the frequency. The transfer function may be represented by a reference analytic expression, e.g. the equation:                               H          ⁡                      (            ω            )                          =                  K          ⁢                                                    (                                  jω                  +                                      𝓏                    1                                                  )                            ⁢                              (                                  ω                  +                                      𝓏                    2                                                  )                            ⁢                                                           ⁢              …              ⁢                                                           ⁢                              (                                  jω                  +                                      𝓏                    m                                                  )                                                                    (                                  jω                  +                                      p                    1                                                  )                            ⁢                              (                                  jω                  +                                      p                    2                                                  )                            ⁢                                                           ⁢              …              ⁢                                                           ⁢                              (                                  jω                  +                                      p                    n                                                  )                                                                        (        1        )            Where (jω+zm) are the zero factors and (jω+pn) are the pole factors of H(ω) and K is called the scale factor. Any finite zero or pole can be repeated. The poles and zeros define the frequency characteristic of a filter uniquely. They may be represented on a pole-zero diagram in the complex plane. Thus, on a pole zero diagram the poles and/or zeros for a particular standard design are located at certain predefined positions depending upon the order of the filter. The transfer function of standard filters is often defined by another type of reference analytic expression e.g. by polynomials N(sn) and D(sn):H(ω)=N(sn)/D(sn)  (2)The frequency response of the filter depends on the coefficients of the polynomials D(sn) and/or N(sn). These coefficients are available from any filter handbook or CAD program and determine the location of the poles (i.e. the roots of D(sn)) and/or the zeros (i.e. the roots of N(sn)) in the s-plane. For example, the poles of a sixth-order Butterworth lowpass filter will lie on a semicircle about the origin in the left half plane and those of a Chebyshev filter will lie on an ellipse. A fifth order Chebyshev pole-zero diagram is shown schematically in FIG. 1.
Filters can be cascaded, that is arranged in series, in order to obtain a response which is the combination of the two filters. In this way a higher order filter may be obtained from a combination of lower order filters. An example, of how to design a cascaded filter arrangement is given in U.S. Pat. No. 5,963,112 the whole contents of which are incorporated by reference. However, when cascading is done, the characteristic of the higher order filter is not the same as when this higher order filter is designed as a single filter. For example, the combination of a cascade of a third and a second order Chebyshev filter does not have the same response as a fifth order Chebyshev filter. In fact, such a cascade has a poorer frequency response especially in the transition bands at the corner frequencies. As shown in FIG. 2, the response of a Butterworth filter in the transition band near the corner frequency changes with order number. For instance, a cascaded combination of a third and a second order Butterworth filter is inferior to a fifth order Butterworth filter as far as the roll-off with frequency above the corner frequency is concerned.
In designing filters, the filter designer is aware that component sensitivity increases with the filter order. The latter should therefore be held as low as possible consistent with the filter specifications. Also higher order passive filters are often bulky. These considerations favour cascading low order filters to obtain a higher order filter. However, as indicated above cascading standard lower order filters does not provide an optimum higher order filter.
Accordingly, a need exists for cascaded higher order filters and a method of designing higher order cascaded filters which provides good frequency response especially in the transition band at the corner frequencies.
A further need exists for higher order filters arrangements and a method of designing higher order filter arrangements which have less bulky components.